Many patents relate to quickly opening and closing screw-and-nut machanism with either the screw or the nut up-and-down typo system. Such patents include U.S. Pat. Nos. 4,834,355, 4,923,185, 5,282,392 and other related patents such as CN 2078210U, CN 1047466A, CN2065969U, CN2057131U, CN1065514A. In these patents, the engagement of thread peaks and thread valleys of the screw and nut is essentially achieved by means of a rigid operation actuating device in the form of a variety of cams, links etc. to directly and rigidly make forced engagement. Problems existed and included failure in engagement of the threads and the threads incapable to work normally. Thus there are two drawbacks: one of the drawbacks is a low percentage of the number of threads engaged relate to the total number of threads. The reason is evident. Since the thread peak of either a triangular thread or trapezoid thread both have a certain width, these thread peaks are likely to collide with each other and not to be engaged when they are engaged in a rigid manner. And the probability of this phenomena happening increases as the width of the thread peak increases. The other drawback is a low percentage of the number of threads engaged in "full thread depth" in the total number of threads. The full thread depth is the depth between the thread peak and thread valley of the semi-nut and screw which can be engaged. In a rigid manner, a complete engagement in "full thread depth" can be attained only when the center of thread peak and the center of thread valley of the semi-nuts and screw are just aligned with each other. It is evident that the probability is also certainly very low, so the most engagements of the threads are not engagement in "full thread depth". Although they are engaged, the engaging depth at every instance is not ideal. Sometimes it begins to work when the engagement of the internal and external threads have not yet reached the specified depth, which lowers loading capacity and reliability of the screw threads. Also in the prior art, there is a Patent CN 2139872Y, having an engaging means for its semi-nuts. Although the present employs a spring for elastically engaging, the disengaging of its semi-nuts is accomplished by another manual operating lever. This patented technique belongs to a technical field of rather primitive, manually operating opening and closing of a screw-and-nut pair without any mechanically programmed, automatically controlled system. It does not belong to the same technical field of the present invention and does not have the function of performing "5 steps for automatically opening and closing". In summary, by using the key technique "mechanically program-controlled, elastically engaging, automatically opening and closing a screw-and-nut pair", the present invention can perform "5 steps for automatically opening and closing", to thereby overcome the various drawbacks which exist in the abovementioned reference documents.